Melbourne

It'll be almost impossible to match the colour - paint ages and changes colour over time. However, colour changes only matter when you get the two colours side by side like in photo. A small difference between the front & back of the door won't be noticeable cos noone will see them side by side. But a splodge in the middle of the door will be very obvious. Basic 'landlord's white' is probably the way to go. Rollering on the paint is quick and very easy if the door edges (and handle & hinges) are masked as per my comment. I would also put a flattened cardboard box or a shit tonne of newspaper down under the door before painting to catch any drips or smears. Acrylic paint is water soluble if you catch it before it dries, so clean up is fairly easy and once done the tape & cardboard and roller etc. can go in the rubbish. Smell will take about 2 days to dissipate. You can speed this up a bit by opening windows if possible, and if not possible by putting a chopped up lemon in a bowl in the room to help mask the smell of the paint.
The really important bit is to get the patch perfectly smooth by sanding it flush. Wrapping a piece of sandpaper round a blocky object (even a couple of dishwashing sponges) will let you sand it flat and smooth. Someone with more sandpaper fu than I have can maybe recommend the size/number of the grit to use.